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Tensor Analysis on Manifolds (Dover Books on Mathematics), by Richard L. Bishop, Samuel I. Goldberg
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"This is a first-rate book and deserves to be widely read." — American Mathematical Monthly
Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject. The authors have treated tensor analysis as a continuation of advanced calculus, striking just the right balance between the formal and abstract approaches to the subject.
The material proceeds from the general to the special. An introductory chapter establishes notation and explains various topics in set theory and topology. Chapters 1 and 2 develop tensor analysis in its function-theoretical and algebraic aspects, respectively. The next two chapters take up vector analysis on manifolds and integration theory. In the last two chapters (5 and 6) several important special structures are studied, those in Chapter 6 illustrating how the previous material can be adapted to clarify the ideas of classical mechanics. The text as a whole offers numerous examples and problems.
A student with a background of advanced calculus and elementary differential equation could readily undertake the study of this book. The more mature the reader is in terms of other mathematical knowledge and experience, the more he will learn from this presentation.
- Sales Rank: #163070 in Books
- Published on: 1980-12-01
- Released on: 1980-12-01
- Original language: English
- Number of items: 1
- Dimensions: 8.27" h x .57" w x 5.62" l, .67 pounds
- Binding: Paperback
- 288 pages
Most helpful customer reviews
13 of 13 people found the following review helpful.
A bit difficult for the non-professional but overall a fascinating book
By magellan
I came to this book with the minimum background--calculus and advanced calculus, differential equations, and some linear algebra, and found it a bit tough going, but still enjoyable. In fact, for me, not being a mathematician but a math hobbyist, really, whose education is mostly in biology and art history, I found it pretty difficult but also quite fascinating and even mind-blowing. I only had the vaguest ideas about tensors, fields, and manifolds before this, although I knew that the theory of manifolds underlies differential geometry and Einstein's famous General Relativity theory.
I understand that the notation in this book is considered old-fashioned and may contribute to the difficulty of reading it. Not having had anything different I don't know if it was harder for me or not, but overall I didn't find the notation too bad. The authors make the interesting point in the introduction that notational developments have occupied much of the work in manifolds, which I found funny. This implies that you can be good at math notation but not that good at the math. So maybe there's hope for me yet. :-)
That issue aside, I found this a very complete and well presented discussion on the subject. Some of it seemed pretty abstract and even counter-intuitive; for example, the concept of distance between two points isn't necessary to have a manifold, and yet having a coordinate neighborhood, or a manifold consisting of differentiable functions is, or other similar properties. It is a little strange to consider that one can perform differentiation on a manifold without the concept of spatial distance, when to my mind taking delta y over delta x at the limit is just shrinking the distance down to nothing in order to obtain the derivative of a function, not to mention that this seems problematic given the requirement of either uniform or non-uniform convergence. How do you know the function converges without some concept of distance? If you're better at this stuff than I am perhaps you could leave me a brief comment if I'm getting something wrong here.
But I still learned a lot, and much of it is pretty amazing and even mind-blowing stuff. People wouldn't need psychedelics if they knew enough to be learning about tensors, manifolds, and topology. They could blow their minds just on this stuff. :-)
So go out and get yourself a book on tensor manifolds and blow your mind the natural way. Higher mathematics is just awesome stuff even if I'm not quite smart enough to really understand it, but I can at least appreciate it, and I probably got a lot further with it than most biology and art history majors. :-)
9 of 10 people found the following review helpful.
Tough for self-study
By Art K.
I have been using this book to study differential geometry for many years - a little bit at a time. This book is a fairly complete introduction to the subject. However, it does a poor job motivating and explaining the subject. I found it necessary to supplement with several other texts to really get a good grasp on the material in the book. A number of times, I have picked up something in another book and have gone back to this book and realized that I had not "gotten it" the first time through. If the book had more examples and concrete calculations it would go a long way to clarifying the material.
I would recommend getting a book like Guggenheimer's Differential Geometry and reading it first. This book then does a good job of generalizing the ideas to many dimensions.
6 of 6 people found the following review helpful.
Can't get over how good this book is.
By E. Ebrahim
As a physics-math major, I have never come across such a perfect book to start differential geometry. I buy a lot of Dover publishing books because of their cheapness, but this one is probably my most valued geometry book. No other book has been this terse and this clear at the same time.
This book provides a solid foundation for everything it does without assuming your understanding of anything before-hand.
For example, physicist-geared introductions tend to hide a lot of the real topology and linear algebra behind the subject. Almost every other book I've read assumes knowledge of dual spaces for vector spaces, and just gives a quick definition. This book does not do any of that. It gives clear mathematical details and motivation to go with them.
Don't let the book's low price make it appear to be low quality.
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